Stopping Set Distributions of Some Reed–Muller Codes
Stopping Set Distributions of Some Reed–Muller Codes
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DOI：
10.1109/tit.2011.2162181
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发表时间：
2011-09
影响因子：
2.5
通讯作者：
Yong Jiang;Shutao Xia;Fang-Wei Fu
中科院分区：
文献类型：
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作者：
Yong Jiang;Shutao Xia;Fang-Wei Fu
使用线性码的停止集和停止集分布来确定该码在二进制擦除信道(BEC)上迭代译码时的性能。设C是具有奇偶校验矩阵H的二进制[n，k]线性码，其中H的行可能是相关的。具有奇偶校验矩阵H的C的停止集S是H的列索引的子集，使得H对S的限制不包含一行权1。停止集分布{Ti(H)}i=0n用奇偶校验矩阵H列举了大小为I的C的停止集的个数。注意，停止集和停止集分布与C的奇偶校验矩阵H有关，设H*是C的奇偶校验矩阵，它是由其对偶码C⊥的所有非零码字组成的。如果一个奇偶校验矩阵H称为BEC最优的，如果Ti(H)=Ti(H*)，i=0，1，…，n，且H的行数最小。本文研究了二元线性码的停止集、停止集分布和BEC-最优奇偶校验矩阵。利用组合学中的有限几何，我们得到了BEC最优奇偶校验矩阵，然后确定了单纯形码、汉明码、一阶Reed-Muller码和扩展的Hamming码的停止集分布，它们是一些Reed-Muller码或它们的缩短或删余形式。
Stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let C be a binary [n,k] linear code with parity-check matrix H, where the rows of H may be dependent. A stopping set S of C with parity-check matrix H is a subset of column indices of H such that the restriction of H to S does not contain a row of weight one. The stopping set distribution {Ti(H)}i=0n enumerates the number of stopping sets with size i of C with parity-check matrix H. Note that stopping sets and stopping set distribution are related to the parity-check matrix H of C. Let H* be the parity-check matrix of C which is formed by all the nonzero codewords of its dual code C⊥. A parity-check matrix H is called BEC-optimal if Ti(H)=Ti(H*), i=0,1,..., n and H has the smallest number of rows. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some Reed-Muller codes or their shortening or puncturing versions.